Given a natural number $q$, let $p = 2^q - 1$ be the $q$-th Mersenne number.
Let $R(q)$ be the minimal square root of $q$ modulo $p$, if one exists. In other words, $R(q)$ is the smallest positive integer $x$ such that $x^2 - q$ is divisible by $p$.
For example, $R(5)=6$ and $R(17)=47569$.
Find $R(74\,207\,281)$. Give your answer modulo $10^9 + 7$.
Note: $2^{74207281}-1$ is prime.
Solution
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